Theorem isrhm 14253

Description: A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Hypotheses

Ref	Expression
isrhm.m	⊢ 𝑀 = (mulGrp‘𝑅)
isrhm.n	⊢ 𝑁 = (mulGrp‘𝑆)

Assertion

Ref	Expression
isrhm	⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))

Proof of Theorem isrhm

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrhm2 14249	. . 3 ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
2	1	elmpocl 6227	. 2 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝑅 ∈ Ring ∧ 𝑆 ∈ Ring))
3		ringgrp 14095	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
4		ringgrp 14095	. . . . . . 7 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ Grp)
5		ghmex 13922	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 𝑆 ∈ Grp) → (𝑅 GrpHom 𝑆) ∈ V)
6	3, 4, 5	syl2an 289	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 GrpHom 𝑆) ∈ V)
7		inex1g 4230	. . . . . 6 ⊢ ((𝑅 GrpHom 𝑆) ∈ V → ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ∈ V)
8	6, 7	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ∈ V)
9		oveq12 6037	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑟 GrpHom 𝑠) = (𝑅 GrpHom 𝑆))
10		fveq2 5648	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
11		fveq2 5648	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (mulGrp‘𝑠) = (mulGrp‘𝑆))
12	10, 11	oveqan12d 6047	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)) = ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))
13	9, 12	ineq12d 3411	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
14	13, 1	ovmpoga 6161	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring ∧ ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ∈ V) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
15	8, 14	mpd3an3 1375	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
16	15	eleq2d 2301	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ 𝐹 ∈ ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))))
17		elin 3392	. . . 4 ⊢ (𝐹 ∈ ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
18		isrhm.m	. . . . . . . 8 ⊢ 𝑀 = (mulGrp‘𝑅)
19		isrhm.n	. . . . . . . 8 ⊢ 𝑁 = (mulGrp‘𝑆)
20	18, 19	oveq12i 6040	. . . . . . 7 ⊢ (𝑀 MndHom 𝑁) = ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))
21	20	eqcomi 2235	. . . . . 6 ⊢ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) = (𝑀 MndHom 𝑁)
22	21	eleq2i 2298	. . . . 5 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)) ↔ 𝐹 ∈ (𝑀 MndHom 𝑁))
23	22	anbi2i 457	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁)))
24	17, 23	bitri 184	. . 3 ⊢ (𝐹 ∈ ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁)))
25	16, 24	bitrdi 196	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))
26	2, 25	biadanii 617	1 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∩ cin 3200  ‘cfv 5333  (class class class)co 6028   MndHom cmhm 13620  Grpcgrp 13663   GrpHom cghm 13907  mulGrpcmgp 14014  Ringcrg 14090   RingHom crh 14245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-mhm 13622  df-grp 13666  df-ghm 13908  df-mgp 14015  df-ur 14054  df-ring 14092  df-rhm 14247

This theorem is referenced by:  rhmmhm  14254  rhmghm  14257  isrhm2d  14260  rhmf1o  14263  rhmco  14269  resrhm  14343  resrhm2b  14344  rhmpropd  14349